(Because—obviously—since the determinant is the product of the eigenvalues,
It’s amazing how rarely people—including textbook authors—actually bother to point this out. (Admittedly, it’s only true over an algebraically closed field such as the complex numbers.) Were you by any chance using Axler?
it serves as a measure of how the transformation distorts area; if the determinant is zero, it means you’ve lost a dimension in the mapping, so you can’t reverse it. But it wouldn’t have been “obvious” if I had only read the Wikipedia article.)
While I certainly agree with the main point of your comment, I nevertheless think that this particular comparison illustrates mainly that the mathematical Wikipedia articles still have a way to go. (Indeed, the property of determinants mentioned above is buried in the middle of the “Further Properties” section of the article, whereas I think it ought to be prominently mentioned in the introduction; in Axler it’s the definition of the determinant [in the complex case]!)
It’s amazing how rarely people—including textbook authors—actually bother to point this out. (Admittedly, it’s only true over an algebraically closed field such as the complex numbers.) Were you by any chance using Axler?
While I certainly agree with the main point of your comment, I nevertheless think that this particular comparison illustrates mainly that the mathematical Wikipedia articles still have a way to go. (Indeed, the property of determinants mentioned above is buried in the middle of the “Further Properties” section of the article, whereas I think it ought to be prominently mentioned in the introduction; in Axler it’s the definition of the determinant [in the complex case]!)
Mostly Bretscher, but checking out Axler’s vicious anti-deteminant screed the other month certainly influenced my comment.